Finitely Suslinian models for planar compacta with applications to Julia sets
Finitely Suslinian models for planar compacta with applications to Julia sets
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DOI：
10.1090/s0002-9939-2012-11607-7
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发表时间：
2010-09
期刊：
影响因子：
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通讯作者：
A. Blokh;Clinton P. Curry;L. Oversteegen
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文献类型：
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作者：
A. Blokh;Clinton P. Curry;L. Oversteegen
紧$X\子集\C$如果与$C\sm X$的无界分支的边界重合，则它是非屏蔽的。称为紧集$X$有限Suslinian，如果每一个直径从零有界的两两不相交的次连续集都是有限的。证明了任一非屏蔽平面紧空间$X$都有一个拓扑唯一的单调映射$m_X：X\to X_{FS}$到有限Suslinian商上，使得$X$到有限Suslinian商上的任何单调映射都通过$m_X$来因子.我们称$(X_{FS}，m_X)$(或者，更宽松地说，$X_{FS}$)为$X$的最优有限Suslinian模型。如果$f：\C\to\C$是分支覆盖映射，$X\子集\C$是完全不变紧的，则$m_X$的适当扩张$M_X$单调半共轭到作为$f$模型的分支覆盖映射$g：\C\to\C$.如果$f$是多项式，$J_f$是它的Julia集，我们证明了$M_X$(或$M_X$)可以单独定义在$J_f$的每个分支$Z$上，作为$Z$到局部连通连续统的最好单调映射.
A compactum $X\subset \C$ is unshielded if it coincides with the boundary of the unbounded component of $\C\sm X$. Call a compactum $X$ finitely Suslinian if every collection of pairwise disjoint subcontinua of $X$ whose diameters are bounded away from zero is finite. We show that any unshielded planar compactum $X$ admits a topologically unique monotone map $m_X:X \to X_{FS}$ onto a finitely Suslinian quotient such that any monotone map of $X$ onto a finitely Suslinian quotient factors through $m_X$. We call the pair $(X_{FS},m_X)$ (or, more loosely, $X_{FS}$) the finest finitely Suslinian model of $X$. If $f:\C\to \C$ is a branched covering map and $X \subset \C$ is a fully invariant compactum, then the appropriate extension $M_X$ of $m_X$ monotonically semiconjugates $f$ to a branched covering map $g:\C\to \C$ which serves as a model for $f$. If $f$ is a polynomial and $J_f$ is its Julia set, we show that $m_X$ (or $M_X$) can be defined on each component $Z$ of $J_f$ individually as the finest monotone map of $Z$ onto a locally connected continuum.